Comment on "First-order methods almost always avoid strict saddle points"

TL;DR

This paper corrects the analysis of Riemannian gradient descent's stability in manifold optimization, providing an explicit step-size bound based on a new retraction L-smooth property, ensuring avoidance of strict saddle points.

Contribution

It corrects previous analysis and introduces a retraction L-smooth property to explicitly bound step-size in Riemannian gradient descent.

Findings

Corrected the stability analysis of Riemannian gradient descent.

Established an explicit step-size bound using retraction L-smoothness.

Ensured avoidance of strict saddle points for almost all initializations.

Abstract

The analysis on the global stability of Riemannian gradient descent method in manifold optimization (i.e., it avoids strict saddle points for almost all initializations) due to Lee et al. (Math. Program. 176:311-337) is corrected. Moreover, an explicit bound on the step-size is presented by the newly introduced retraction L-smooth property.